Let us recall now a few basic definitions related to manifolds
with corners [1]}, [2]}, [3]}, [4]}, [5]}.
Recall that a manifold with corners \(M\)  of dimension \(n\)  is a topological
space locally modeled on \(\mathbb {R}^n_k := [0, \infty )^{k} \times \mathbb {R}^{n-k}\)  with
smooth transition functions. Each of the spaces introduced in
(REF ) provides an
example of a manifold with corners. Given a point \(x\)  in a manifold with corners
\(M\) , the depth of \(x\)  in \(M\)  is defined locally using Definition
REF . More precisely, the depth of \(x\)  in \(M\)  is
the number of non-negative coordinate
functions vanishing at \(p\)  in any local coordinate chart at \(p\) . It is the
least \(k\)  such that there exists a chart near \(U\)  with values in \(\mathbb {R}_k^n\) .
Let \((M)_k\)  be the set of points of \(M\)  of depth \(k\) . Its connected components are called
the open boundary faces of codimension (or depth) \(k\)  of \(M\) . A boundary face of depth \(k\) 
is the closure of an open boundary face of depth \(k\) . A boundary face \(H\)  of codimension
one of \(M\)  will be called a boundary hyperface of \(M\) . The union of
the boundary hyperfaces \(H\)  of \(M\)  is denoted \(\partial M\) ; it is
the boundary of \(M\) . If \(H\)  is a boundary hyperface of \(M\)  or
a disjoint union of boundary hyperfaces of \(M\)  and \(x_H\)  is
a smooth function on \(M\)  such that \(H = \lbrace x_H = 0\rbrace \)  and \(dx_H\ne 0\)  on \(H\) 
(i.e. it is non-degenerate everywhere on \(H\) ), then \(x_H\)  is called a boundary
defining function of \(H\) .
